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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 860 · 40 questions
TalentJR

How to solve it (Use the identity, skip the guessing): (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b²; so a² + b² = (a + b)² − 2ab and (a − b)² = (a + b)² − 4ab. For x² − Sx + P = 0: the sum of the roots is S and the product is P. To factorise x² + Sx + P, find two numbers that multiply to P and add up to S.

NameDateTime takenScore ___ / 40
  1. 1.If a + b = 3 and ab = 2, what is a² + b²? (a) 9 (b) 7 (c) 5 (d) 13
  2. 2.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 8)(x + 9) (c) (x + 1)(x + 72) (d) (x + 1)(x + 16)
  3. 3.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 56) (c) (x − 7)(x − 8) (d) (x + 1)(x + 14)
  4. 4.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3601 (b) 3721 (c) 3731 (d) 3481
  5. 5.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 36) (b) (x + 1)(x + 12) (c) (x − 4)(x − 9) (d) (x + 4)(x + 9)
  6. 6.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8099 (b) 7931 (c) 7921 (d) 8281
  7. 7.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x + 1)(x + 10) (c) (x + 4)(x + 7) (d) (x − 4)(x − 7)
  8. 8.If a + b = 14 and ab = 24, what is a² + b²? (a) 244 (b) 148 (c) 172 (d) 196
  9. 9.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 15 (c) 16 (d) 17
  10. 10.If a + b = 12 and ab = 35, what is (a − b)²? (a) 144 (b) 2 (c) 4 (d) 74
  11. 11.What is the sum of the roots of x² − 7x + 12 = 0? (a) 6 (b) 8 (c) 12 (d) 7
  12. 12.Which is the correct factorisation of x² + 12x + 27? (a) (x + 3)(x + 9) (b) (x − 3)(x − 9) (c) (x + 1)(x + 11) (d) (x + 1)(x + 27)
  13. 13.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 7)(x + 9) (c) (x − 7)(x − 9) (d) (x + 1)(x + 15)
  14. 14.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x + 1)(x + 8) (c) (x − 2)(x − 4) (d) (x + 2)(x + 4)
  15. 15.If a + b = 16 and ab = 48, what is a² + b²? (a) 352 (b) 208 (c) 256 (d) 160
  16. 16.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2516 (b) 2916 (c) 2116 (d) 2926
  17. 17.What are the roots of x² − 9x + 14 = 0? (a) 3 and 6 (b) 2 and 7 (c) 1 and 8 (d) 1 and 14
  18. 18.If a + b = 12 and ab = 32, what is a² + b²? (a) 112 (b) 144 (c) 80 (d) 208
  19. 19.If a + b = 15 and ab = 54, what is (a − b)²? (a) 225 (b) 9 (c) 3 (d) 117
  20. 20.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 4909 (b) 4489 (c) 5329 (d) 5339
  21. 21.If a + b = 7 and ab = 12, what is a² + b²? (a) 73 (b) 49 (c) 25 (d) 37
  22. 22.What is the sum of the roots of x² − 6x + 8 = 0? (a) 8 (b) 5 (c) 6 (d) 7
  23. 23.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8104 (b) 8474 (c) 8464 (d) 7744
  24. 24.If a + b = 17 and ab = 72, what is (a − b)²? (a) 217 (b) 145 (c) 289 (d) 1
  25. 25.If a + b = 11 and ab = 18, what is a² + b²? (a) 157 (b) 103 (c) 121 (d) 85
  26. 26.What is the product of the roots of x² − 10x + 21 = 0? (a) 22 (b) 21 (c) 20 (d) 10
  27. 27.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 484 (b) 494 (c) 324 (d) 404
  28. 28.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x + 1)(x + 35) (c) (x + 5)(x + 7) (d) (x − 5)(x − 7)
  29. 29.What are the roots of x² − 5x + 6 = 0? (a) 2 and 4 (b) 1 and 6 (c) 1 and 4 (d) 2 and 3
  30. 30.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 676 (b) 1156 (c) 686 (d) 884
  31. 31.If a + b = 11 and ab = 18, what is (a − b)²? (a) 7 (b) 121 (c) 85 (d) 49
  32. 32.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 396 (b) 334 (c) 484 (d) 324
  33. 33.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 961 (b) 841 (c) 851 (d) 899
  34. 34.What is the sum of the roots of x² − 11x + 24 = 0? (a) 24 (b) 10 (c) 11 (d) 12
  35. 35.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4366 (b) 4884 (c) 5476 (d) 4356
  36. 36.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x − 5)(x − 8) (c) (x + 1)(x + 12) (d) (x + 5)(x + 8)
  37. 37.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 16 (d) 17
  38. 38.What is the product of the roots of x² − 6x + 8 = 0? (a) 6 (b) 9 (c) 8 (d) 7
  39. 39.If a + b = 17 and ab = 72, what is a² + b²? (a) 289 (b) 217 (c) 145 (d) 433
  40. 40.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1946 (b) 1936 (c) 1296 (d) 1616

Answer key

  1. 5
  2. (x + 8)(x + 9)
  3. (x + 7)(x + 8)
  4. 3721
  5. (x + 4)(x + 9)
  6. 7921
  7. (x + 4)(x + 7)
  8. 148
  9. 16
  10. 4
  11. 7
  12. (x + 3)(x + 9)
  13. (x + 7)(x + 9)
  14. (x + 2)(x + 4)
  15. 160
  16. 2916
  17. 2 and 7
  18. 80
  19. 9
  20. 5329
  21. 25
  22. 6
  23. 8464
  24. 1
  25. 85
  26. 21
  27. 484
  28. (x + 5)(x + 7)
  29. 2 and 3
  30. 676
  31. 49
  32. 324
  33. 841
  34. 11
  35. 4356
  36. (x + 5)(x + 8)
  37. 17
  38. 8
  39. 145
  40. 1936

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